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Magazine Chapter 12: Problem 8
Graph each exponential function. $$ y=\left(\frac{1}{3}\right)^{x}+2 $$
Short Answer
Expert verified
Sketch an exponential decay curve approaching \( y = 2 \) from above.
Step by step solution
01
Understand the Basic Form
We start by identifying the components of the function. The given function is \[ y = \left( \frac{1}{3} \right)^{x} + 2 \]This is an exponential function where \( a = \frac{1}{3} \) and there is a vertical shift upwards by 2 units.
02
Identify Key Points
Choose several values for \( x \) to calculate corresponding \( y \) values. Common points include when \( x = 0, 1, -1 \):- For \( x = 0 \), \( y = \left( \frac{1}{3} \right)^{0} + 2 = 1 + 2 = 3 \).- For \( x = 1 \), \( y = \left( \frac{1}{3} \right)^{1} + 2 = \frac{1}{3} + 2 = \frac{7}{3} \) or approximately 2.33.- For \( x = -1 \), \( y = \left( \frac{1}{3} \right)^{-1} + 2 = 3 + 2 = 5 \).
03
Determine Asymptote and Behavior
The horizontal asymptote is the line that the graph approaches but never touches or crosses. For our function, as \( x \to \infty \), \( \left( \frac{1}{3} \right)^x \to 0 \), so the horizontal asymptote is \( y = 2 \).Also, as \( x \to -\infty \), \( \left( \frac{1}{3} \right)^x \) grows very large, thus \( y \to \infty \).
04
Plot the Points and Asymptote
Plot the points found in Step 2 and draw the horizontal asymptote at \( y = 2 \). The graph should show exponential decay towards the asymptote as \( x \to \infty \) and rise steeply as \( x \to -\infty \).
05
Draw the Curve
Using the points and understanding from previous steps, sketch a smooth curve that passes through the plotted points and approaches the asymptote \( y = 2 \) from above.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Exponential Functions
Graphing an exponential function involves translating the algebraic expression into a visual format on a coordinate plane. The exponential function we are dealing with is \[ y = \left( \frac{1}{3} \right)^{x} + 2 \]When graphing these types of functions, it's important to recognize the components: the base, in this case \( \frac{1}{3} \), and any vertical shifts, like the \(+2\) here, which moves the entire graph upwards by 2 units.
- Start by choosing a range of x-values. Useful points often include \( x = 0, 1, ext{and} -1 \).
- For \( x = 0\), \( y = 3 \).
- For \( x = 1 \), \( y = \frac{7}{3} \).
- For \( x = -1 \), \( y = 5 \).
Asymptotes in Graphs
Asymptotes are essential when graphing exponential functions. They help define the boundaries that the graph will never cross. For the function \( y = \left( \frac{1}{3} \right)^{x} + 2 \), the horizontal asymptote is at \( y = 2 \). This is due to the constant \(+2\), shifting the graph upwards.
As \( x \to \infty \), the expression \( \left( \frac{1}{3} \right)^{x} \) approaches zero. Therefore, the graph hovers closer and closer to \( y = 2 \), never actually touching it. However, as \( x \to -\infty \), it grows very large because negative exponents invert the fraction.
As \( x \to \infty \), the expression \( \left( \frac{1}{3} \right)^{x} \) approaches zero. Therefore, the graph hovers closer and closer to \( y = 2 \), never actually touching it. However, as \( x \to -\infty \), it grows very large because negative exponents invert the fraction.
- The asymptote acts as a visual guideline, showing where the function stabilizes.
- It forms a boundary for the graph's behavior.
- Understanding asymptotes helps predict and model the graph's progression over different x-values.
Exponential Decay
Exponential decay describes how quantities decrease rapidly and then more slowly over time or space. In the function \( y = \left( \frac{1}{3} \right)^{x} + 2 \), the decay is evident as the base \( \frac{1}{3} \) is a fraction less than one.
- As \( x \) increases, the expression raises a small fraction to a power, causing rapid shrinkage towards zero.
- This is why the graph of \( \left( \frac{1}{3} \right)^{x} \) declines as \( x \to \infty \), emphasizing the decay.
- All exponential decay functions approach a horizontal asymptote, which, in this case, is \( y = 2 \).
